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Categorical local Langlands for $\mathrm{GL}_n$ for parameters of Langlands-Shahidi type with integral coefficients

Konrad Zou

TL;DR

This work realizes a categorical geometrization for GL$_n$ with Langlands–Shahidi type $L$-parameters by restricting to the Langlands–Shahidi locus ${\rm LSt}$ and employing the spectral action of $\Ind\Perf({\rm Par}_{\hat G})$. It proves an $ ext{Ind}\Perf({\rm LSt})$-linear, $t$-exact equivalence between $\Ind\Perf^\mathrm{qc}({\rm LSt})$ with nilpotent support and the derived category $\mathcal{D}^{\rm LSt}(\text{Bun}_G)$, preserving compacts and sending the structure sheaf to the Whittaker object, thereby coupling coherent and automorphic worlds for a broad class of $L$-parameters. The authors develop explicit Hecke operator computations at Langlands–Shahidi type parameters, use Hodge–Newton reducibility to control stalks, and derive a full decomposition into isotypic components indexed by $X^*(S_{\varphi})$; these ingredients yield a full proof of the conjecture in characteristic zero and, via reductions and base changes, extend the results to torsion and integral coefficients. Applications include torsion-vanishing statements for Lubin–Tate cohomology and PEL-type Shimura varieties of type A, as well as the construction of perverse Hecke eigensheaves with Langlands–Shahidi eigenvalues.

Abstract

We prove the categorical form of Fargues' geometrization conjecture for $\mathrm{GL}_n$ along $L$-parameters of Langlands-Shahidi type for rational, torsion, and integral coefficients. Additionally, we prove that in this case the categorical equivalence is $t$-exact, which yields new torsion vanishing results in the cohomology of unitary Shimura varieties.

Categorical local Langlands for $\mathrm{GL}_n$ for parameters of Langlands-Shahidi type with integral coefficients

TL;DR

This work realizes a categorical geometrization for GL with Langlands–Shahidi type -parameters by restricting to the Langlands–Shahidi locus and employing the spectral action of . It proves an -linear, -exact equivalence between with nilpotent support and the derived category , preserving compacts and sending the structure sheaf to the Whittaker object, thereby coupling coherent and automorphic worlds for a broad class of -parameters. The authors develop explicit Hecke operator computations at Langlands–Shahidi type parameters, use Hodge–Newton reducibility to control stalks, and derive a full decomposition into isotypic components indexed by ; these ingredients yield a full proof of the conjecture in characteristic zero and, via reductions and base changes, extend the results to torsion and integral coefficients. Applications include torsion-vanishing statements for Lubin–Tate cohomology and PEL-type Shimura varieties of type A, as well as the construction of perverse Hecke eigensheaves with Langlands–Shahidi eigenvalues.

Abstract

We prove the categorical form of Fargues' geometrization conjecture for along -parameters of Langlands-Shahidi type for rational, torsion, and integral coefficients. Additionally, we prove that in this case the categorical equivalence is -exact, which yields new torsion vanishing results in the cohomology of unitary Shimura varieties.

Paper Structure

This paper contains 8 sections, 69 theorems, 93 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Parameters of Langlands-Shahidi type for GLn
  3. Reduction to generic local Langlands in families
  4. Irreducible case of the conjecture
  5. Explicit computation of Hecke operators at parameters of Langlands-Shahidi type
  6. Proof of the conjecture
  7. t-exactness
  8. Applications

Key Result

Theorem 1.3

Let ${\mathrm{LSt}}$ be the locus of $\mathop{\mathrm{Par}}\nolimits_{\widehat{G}}$ consisting of $L$-parameters of Langlands-Shahidi type. There is an $\mathop{\mathrm{Ind}}\limits\mathop{\mathrm{Perf}}\nolimits({\mathrm{LSt}})$-linear equivalence preserving compact objects, mapping $\mathcal{O}$ to $\mathcal{W}^{\mathrm{LSt}}$.

Figures (3)

  • Figure 1: $b_{(2,2,1),L}$ with the smallest slope marked orange and a modification of the smallest slope
  • Figure 2: $b_{(1,2,3),L}$ and $b_{(2,2,1),L}$, the dotted line shows that the modification is $\omega$-Hodge-Newton reducible for $\mathrm{GL}_2\times\mathrm{GL}_6$
  • Figure 3: Analysis of a modification of $b_{(2,1),L}$ as above. Observe that the red slope is bigger than the biggest black slope, and the blue dashed slope is bigger than the orange one. The dashed line is $b_{(3,1),L}$, and this is non-reduciblerelative to $b_{(2,1),L}$ and we see that it is neither Hodge-Newton nor $\omega$-Hodge-Newton reducible.

Theorems & Definitions (166)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 1.3: \ref{['thm: categorical equivalence']}
  • Theorem 1.4: \ref{['thm: t-exact']}
  • Theorem 1.5: special case of \ref{['cor: explict hecke * stalk']}
  • Remark 1.6
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • ...and 156 more